Hamiltonian Dynamics Hamilton s equations The system evolves following aH H q dz Alternative expression JVH(), where dt d×d d×d d×d 0 dxd For HMC, h(q,p)=u(a)+k(p, k(p)=p M-lp/2 qi =[m pli aU =agiHamiltonian Dynamics • Hamilton’s Equations – The system evolves following: 𝑞ሶ 𝑖 = 𝜕𝐻 𝜕𝑝 𝑝ሶ 𝑖 = − 𝜕𝐻 𝜕𝑞 – Alternative expression: 𝑑𝑧 𝑑𝑡 = 𝐽𝛻𝐻 𝑧 , where 𝐽 = 0𝑑×𝑑 𝐼𝑑×𝑑 −𝐼𝑑×𝑑 0𝑑×𝑑 – For HMC, 𝐻 𝑞, 𝑝 = 𝑈 𝑞 + 𝐾 𝑝 ,𝐾 𝑝 = 𝑝 𝑇𝑀−1𝑝/2, 𝑞ሶ 𝑖 = 𝑀−1𝑝 𝑖 𝑝ሶ 𝑖 = − 𝜕𝑈 𝜕𝑞𝑖